6.1 Q-8

Question Statement

We are given two circles with the following equations:

1. $x^2 + y^2 + 2x - 8 = 0$
2. $x^2 + y^2 - 6x + 6y - 46 = 0$

We need to show that these two circles touch internally.

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Background and Explanation

To determine if two circles touch internally, we need to check the following condition:

• The distance between their centers should be equal to the difference in their radii.

Key Concepts:

1. The center of a circle in the form $x^2 + y^2 + 2gx + 2fy + c = 0$ is $(-g, -f)$.
2. The radius of a circle is given by: $r = \sqrt{g^2 + f^2 - c}$

We will use these formulas to find the centers and radii of both circles, then check if the distance between their centers equals the difference in their radii.

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Solution

Step 1: Extract the centers and radii of both circles

Circle (1): $x^2 + y^2 + 2x - 8 = 0$

• The equation is in the general form $x^2 + y^2 + 2gx + 2fy + c = 0$, where:
  • $g = 1$, $f = 0$, and $c = -8$.
• The center of the circle is $(-g, -f) = (-1, 0)$.
• The radius is:

$$r_1 = \sqrt{g^2 + f^2 - c} = \sqrt{1^2 + 0^2 - (-8)} = \sqrt{1 + 8} = 3$$

Circle (2): $x^2 + y^2 - 6x + 6y - 46 = 0$

• The equation is in the form $x^2 + y^2 + 2gx + 2fy + c = 0$, where:
  • $g = -3$, $f = 3$, and $c = -46$.
• The center of the circle is $(-g, -f) = (3, -3)$.
• The radius is:

$$r_2 = \sqrt{g^2 + f^2 - c} = \sqrt{(-3)^2 + 3^2 - (-46)} = \sqrt{9 + 9 + 46} = \sqrt{64} = 8$$

Step 2: Calculate the distance between the centers

The distance $d$ between the centers $(-1, 0)$ and $(3, -3)$ is given by the distance formula:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Substitute the coordinates of the centers:

$$d = \sqrt{(3 - (-1))^2 + (-3 - 0)^2} = \sqrt{(3 + 1)^2 + (-3)^2} = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5$$

Step 3: Check the condition for internal tangency

For the circles to touch internally, the distance between the centers should be equal to the difference in their radii:

$$r_2 - r_1 = 8 - 3 = 5$$

Since the distance between the centers is 5, which is equal to the difference in the radii, we can conclude that the circles touch internally.

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Key Formulas or Methods Used

• Center of a Circle:
  From the general equation $x^2 + y^2 + 2gx + 2fy + c = 0$, the center is $(-g, -f)$.

• Radius of a Circle:
  The radius is given by: $r = \sqrt{g^2 + f^2 - c}$

• Distance Between Two Points:
  The distance between the centers $(x_1, y_1)$ and $(x_2, y_2)$ is: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

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Summary of Steps

1. Step 1: Find the centers and radii of both circles.
2. Step 2: Calculate the distance between the centers of the two circles.
3. Step 3: Check if the difference in the radii is equal to the distance between the centers.
4. Step 4: Conclude that the circles touch internally because the difference in the radii equals the distance between their centers.